Special and general types
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Recall from the discussion at cohomology that every notion of cohomology (e.g. group cohomology, abelian sheaf cohomology, etc) is given by Hom-spaces in an (∞,1)-topos . Cohomology on an object with coefficients in an object is
The cup product is an operation on cocycles with coefficients and that is induced from a pairing morphism
in . In applications this is often a pairing operation with , i.e. , and typically it is the product morphism for a ring object structure on the coefficients .
If and are two cocycles in and , respectively, then their cup product with respect to this pairing is the cocycle
Via the Dold-Kan correspondence
When the coefficient object ∞Grpd is “sufficiently abelian” in that under the Dold-Kan correspondence it is represented by a chain complex then using the lax monoidalness of the Dold-Kan correspondence (see at monoidal Dold-Kan correspondence) one obtains a chain complex model for the cup product which makes the origin of the typical grading shift manifest.
Using all this, then for
a given chain map, then this induces a map of the corresponding Kan complexes
as the following composite
With this in hand then for any homotopy type, the cup product on its cohomology with coefficients in and is induced by just homming into this morphism:
For example if
is the chain complex concentrated in degree and , respectively, on the group of integers, then
is the corresponding Eilenberg-MacLane space which classifies ordinary cohomology (singular cohomology) with integral coefficients in the given degree. By the nature of the tensor product of chain complexes one has
Hence we may take and and we get a cup product
On Moore complexes of cosimplicial algebras
For any cosimplicial algebra, its dual Moor cochain complex naturally inherits the structure of a dg-algebra under the cup product.
The general formula is literally the same as that for the case where is functions on the singular complex of a space, which is discussed below. For the moment, see below.
This cup product operation on is not in general commutative. However, it is a standard fact that it becomes commutative after passing to cochain cohomology.
This suggests that the cup product should be, while not commutative, homotopy commutative in that it makes a homotopy commutative monoid object.
This in turn should mean that is an algebra over an operad for the E-∞ operad.
That this is indeed the case is the main statement in (Berger-Fresse 01)
In singular cohomology
A special case of the cup product on Moore complexes is the complex of singular cohomology, which is the Moore complex of the cosimplicial algebra of functions on the singular simplicial set of a topological space.
Often in the literature by cup product is meant specifically the realization of the cup product on singular cohomology.
For a topological space, let be the simplicial set of -simplices in – the fundamental ∞-groupoid of .
For some ring, let be the cosimplicial ring of -valued functions on the spaces of -simplices. The corresponding Moore cochain complex is the cochain complex whose cochain cohomology is the singular cohomology of the space : a homogeneous element is a function on -simplices in .
Write, as usual, for , for the totally ordered set with elements. For an injective order preserving map and some cosimplicial object, write for the image of this map under .
Specifically, for let be the map that sends to and let be the map that sends to .
Then the cup product
is the cochain map that on homogeneous elements is defined by the formula
There is some glue missing here to connect this back to the above general definition, something involving the Eilenberg-Zilber map.
This means that .
This cup product enjoys the following properties:
it is indeed a cochain complex morphism as claimed, in that it respects the differential: for any homogeneous as above we have
the image of the cup product on cochain cohomology
is associative and distributes over the addition in .
Accordingly, the cup product makes into a ring: the cohomology ring on the ordinary cohomology of .
See for instance section 3.2 of
- Hatcher, Algebraic Topology (web pdf)
In abelian sheaf cohomology
Traditionally the cup product is considered for abelian cohomology, such as generalized (Eilenberg-Steenrod) cohomology and more generally abelian sheaf cohomology.
In that case all coefficient objects are complexes of sheaves and the pairing that one usually considers is the tensor product of chain complexes
In abelian Čech cohomology
The cup product has a simple expression in abelian Čech cohomology.
For and two chain complexes (of sheaves of abelian groups) construct a morphism of Čech complexes
by sending and to
For instance (Brylinski, section (1.3)) spring
In Čech-Deligne cohomology (ordinary differential cohomology)
For the case that of Čech hypercohomology with coefficients in Deligne complexes the above yields the Beilinson-Deligne cup-product for ordinary differential cohomology.
The cup product in Čech cohomology is discussed for instance in section 1.3 of
Recall from the discussion at models for ∞-stack (∞,1)-toposes that all hypercomplete ∞-stack (∞,1)-toposes are modeled by the model structure on simplicial presheaves. Accordingly understanding the cup product on simplicial presheaves goes a long way towards the most general description. For a bit of discussion of this see around page 19 of
An early treatment of cup product can be found in this classic
- Whitney, On Products in a Complex (JSTOR)